diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index 0d3912847dca0..e9b6d53a095d3 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -580,6 +580,7 @@ Other
 
 - Bug in :meth:`DataFrame.replace` and :meth:`Series.replace` incorrectly raising ``AssertionError`` instead of ``ValueError`` when invalid parameter combinations are passed (:issue:`36045`)
 - Bug in :meth:`DataFrame.replace` and :meth:`Series.replace` with numeric values and string ``to_replace`` (:issue:`34789`)
+- Bug in :meth:`DataFrame.replace` and :meth:`Series.replace` incorrectly casting from ``PeriodDtype`` to object dtype (:issue:`34871`)
 - Fixed bug in metadata propagation incorrectly copying DataFrame columns as metadata when the column name overlaps with the metadata name (:issue:`37037`)
 - Fixed metadata propagation in the :class:`Series.dt`, :class:`Series.str` accessors, :class:`DataFrame.duplicated`, :class:`DataFrame.stack`, :class:`DataFrame.unstack`, :class:`DataFrame.pivot`, :class:`DataFrame.append`, :class:`DataFrame.diff`, :class:`DataFrame.applymap` and :class:`DataFrame.update` methods (:issue:`28283`) (:issue:`37381`)
 - Bug in :meth:`Index.union` behaving differently depending on whether operand is a :class:`Index` or other list-like (:issue:`36384`)
diff --git a/pandas/core/internals/blocks.py b/pandas/core/internals/blocks.py
index 13e428c0419ec..c20afb06cc1a2 100644
--- a/pandas/core/internals/blocks.py
+++ b/pandas/core/internals/blocks.py
@@ -752,36 +752,21 @@ def replace(
             to_replace = convert_scalar_for_putitemlike(to_replace, values.dtype)
 
         mask = missing.mask_missing(values, to_replace)
+        if not mask.any():
+            # Note: we get here with test_replace_extension_other incorrectly
+            #  bc _can_hold_element is incorrect.
+            return [self] if inplace else [self.copy()]
 
-        try:
-            blocks = self.putmask(mask, value, inplace=inplace)
-            # Note: it is _not_ the case that self._can_hold_element(value)
-            #  is always true at this point.  In particular, that can fail
-            #  for:
-            #   "2u" with bool-dtype, float-dtype
-            #   0.5 with int64-dtype
-            #   np.nan with int64-dtype
-        except (TypeError, ValueError):
-            # GH 22083, TypeError or ValueError occurred within error handling
-            # causes infinite loop. Cast and retry only if not objectblock.
-            if is_object_dtype(self):
-                raise
-
-            if not self.is_extension:
-                # TODO: https://github.com/pandas-dev/pandas/issues/32586
-                # Need an ExtensionArray._can_hold_element to indicate whether
-                # a scalar value can be placed in the array.
-                assert not self._can_hold_element(value), value
-
-            # try again with a compatible block
-            block = self.astype(object)
-            return block.replace(
+        if not self._can_hold_element(value):
+            blk = self.astype(object)
+            return blk.replace(
                 to_replace=original_to_replace,
                 value=value,
-                inplace=inplace,
+                inplace=True,
                 regex=regex,
             )
 
+        blocks = self.putmask(mask, value, inplace=inplace)
         blocks = extend_blocks(
             [b.convert(numeric=False, copy=not inplace) for b in blocks]
         )
diff --git a/pandas/tests/frame/methods/test_replace.py b/pandas/tests/frame/methods/test_replace.py
index 9011136e12afb..8f3dcc96ddc3d 100644
--- a/pandas/tests/frame/methods/test_replace.py
+++ b/pandas/tests/frame/methods/test_replace.py
@@ -1523,18 +1523,18 @@ def test_replace_with_duplicate_columns(self, replacement):
 
         tm.assert_frame_equal(result, expected)
 
-    @pytest.mark.xfail(
-        reason="replace() changes dtype from period to object, see GH34871", strict=True
-    )
-    def test_replace_period_ignore_float(self):
+    def test_replace_period_ignore_float(self, frame_or_series):
         """
         Regression test for GH#34871: if df.replace(1.0, 0.0) is called on a df
         with a Period column the old, faulty behavior is to raise TypeError.
         """
-        df = DataFrame({"Per": [pd.Period("2020-01")] * 3})
-        result = df.replace(1.0, 0.0)
-        expected = DataFrame({"Per": [pd.Period("2020-01")] * 3})
-        tm.assert_frame_equal(expected, result)
+        obj = DataFrame({"Per": [pd.Period("2020-01")] * 3})
+        if frame_or_series is not DataFrame:
+            obj = obj["Per"]
+
+        expected = obj.copy()
+        result = obj.replace(1.0, 0.0)
+        tm.assert_equal(expected, result)
 
     def test_replace_value_category_type(self):
         """
diff --git a/pandas/tests/series/methods/test_replace.py b/pandas/tests/series/methods/test_replace.py
index 27cf15b4e187b..565debb98d8cc 100644
--- a/pandas/tests/series/methods/test_replace.py
+++ b/pandas/tests/series/methods/test_replace.py
@@ -424,10 +424,12 @@ def test_replace_only_one_dictlike_arg(self):
         with pytest.raises(ValueError, match=msg):
             ser.replace(to_replace, value)
 
-    def test_replace_extension_other(self):
+    def test_replace_extension_other(self, frame_or_series):
         # https://github.com/pandas-dev/pandas/issues/34530
-        ser = pd.Series(pd.array([1, 2, 3], dtype="Int64"))
-        ser.replace("", "")  # no exception
+        obj = frame_or_series(pd.array([1, 2, 3], dtype="Int64"))
+        result = obj.replace("", "")  # no exception
+        # should not have changed dtype
+        tm.assert_equal(obj, result)
 
     def test_replace_with_compiled_regex(self):
         # https://github.com/pandas-dev/pandas/issues/35680